Scoring... Conclusion

We have come up with a two-way bet in which both parties have incentive to reveal exactly their beliefs regarding the true probability of an event. Neither expects to lose, at worst breaking even— and then only when they agree regarding the true probability. It turns out that all these properties still hold when multiple players are involved. We simply have each player bet against every player. Let $p_i$ be the probability declared by player $i$ of $n$. Then let $$ U=\frac{1}{n}\sum_{i=1}{\lg{p_i}} $$ and $$ V=\frac{1}{n}\sum_{i=1}{\lg{\!\left(1-p_i\right)}} $$ We can set the payout to player $j$ to $\lg{p_j}-U$ if the event occurs, and $\lg{\!\left(1-p_j\right)}-V$ if it does not. Note that the total of all $n$ payouts is always zero.

And this is how the Game of Game of Thrones is scored. The score for the four-way Arya “Dead Pool” at the end works basically the same way, but with four possible outcomes rather than just two. The average score will always come to zero, but every player expects that they will do better than everyone else on any unresolved questions. But most importantly, the winner will be the player (ostensibly) least surprised by the events on the show.

A final detail on scoring. Since we in the GoGoT are not actually paying each other based on scores, the real objective is strictly not to maximize score, but merely to have the best score in the group. This might have made it worthwhile for someone to lie a little about what they thought the probabilities were. If you figured that out, then kudos to you.